open import 1Lab.Reflection.Record

open import Cat.Instances.Functor
open import Cat.Instances.Product
open import Cat.Diagram.Product
open import Cat.Prelude
open import Cat.Strict
module Cat.Instances.StrictCat where
open Product
open is-product
open Precategory
open Functor

private variable
  o h : Level

The category of strict categoriesπŸ”—

Recall that a precategory is said strict if its space of objects is a Set. While general precategories are too homotopically interesting to fit into a Precategory (because functor spaces will not, in general, be h-sets), the strict categories do form a precategory, which we denote 

private unquoteDecl eqv = declare-record-iso eqv (quote Functor)

Functor-is-set : βˆ€ {o h} {C D : Precategory o h} β†’ is-set (Ob D)
               β†’ is-set (Functor C D)
Functor-is-set {o = o} {h} {C} {D} dobset = Iso→is-hlevel! 2 eqv where instance
  Dob : H-Level (Ob D) 2
  Dob = basic-instance 2 dobset
Strict-cats : βˆ€ o h β†’ Precategory _ _
Strict-cats o h .Ob = Σ[ C ∈ Precategory o h ] (is-strict C)
Strict-cats o h .Hom (C , _) (D , _) = Functor C D
Strict-cats o h .id  = Id
Strict-cats o h ._∘_ = _F∘_
Strict-cats o h .idr _       = Functor-path (Ξ» _ β†’ refl) Ξ» _ β†’ refl
Strict-cats o h .idl _       = Functor-path (Ξ» _ β†’ refl) Ξ» _ β†’ refl
Strict-cats o h .assoc _ _ _ = Functor-path (Ξ» _ β†’ refl) Ξ» _ β†’ refl

This assembles into a Precategory because the only bit of a Functor that doesn’t have a fixed h-level is the object mapping; By asking that D be a strict category, this fixes the functors to be sets.

Strict-cats o h .Hom-set _ (D , dset) = Functor-is-set dset

ProductsπŸ”—

We prove that Strict-cats has products. This is because is and h-levels are closed under products.

Strict-cats-products
  : {C D : Precategory o h}
  β†’ (cob : is-set (Ob C)) (dob : is-set (Ob D))
  β†’ Product (Strict-cats o h) (C , cob) (D , dob)
Strict-cats-products {C = C} {D = D} cob dob = prod where
  prod : Product (Strict-cats _ _) (C , cob) (D , dob)
  prod .apex = C Γ—αΆœ D , Γ—-is-hlevel 2 cob dob
  prod .π₁ = Fst {C = C} {D = D}
  prod .Ο€β‚‚ = Snd {C = C} {D = D}
  prod .has-is-product .⟨_,_⟩ p q = Cat⟨ p , q ⟩
  prod .has-is-product .Ο€β‚βˆ˜βŸ¨βŸ© = Functor-path (Ξ» _ β†’ refl) Ξ» _ β†’ refl
  prod .has-is-product .Ο€β‚‚βˆ˜βŸ¨βŸ© = Functor-path (Ξ» _ β†’ refl) Ξ» _ β†’ refl
  prod .has-is-product .unique p q =
    Functor-path (Ξ» x i β†’ Fβ‚€ (p i) x , Fβ‚€ (q i) x) Ξ» f i β†’ F₁ (p i) f , F₁ (q i) f